Integral Inequality Constraints

Consider the optimal control problem in the last section, but with integral equality constraints replaced with the inequality constraints 

In this case. Equation (6.18) provides the augmented functional M whose variation is given by Equation (6.20). From the John Multiplier Theorem (Section 4.5.1, p. 113), the necessary conditions for the minimum are 

The integral constraints are equivalent to the differential equations governing the additional state variables as follows  

Let us consider Example 6.10 with its equality constraint replaced with 

In other words, we want distillate purity greater than or equal to y. Then the necessary conditions for the minimum are 

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Consider the batch reactor problem in Example 7.1 (p. 192) subject to the following integral inequality constraints ... 

Figure 7.19 The optimal states when the integral inequality constraints are satisfied upon convergence in Example 7.8...

In this final chapter, we study the multisite refinery and petrochemical integration problem, explained in Chapter 5, under uncertainty. The randomness considered includes both the objective function and right-hand side parameters of inequality constraints. As pointed out in the previous chapters, considering such strategic planning decisions requires proper handling of uncertainties as they play a maj or role in the final decision making process. 

Table 1 presents the description of the decision variables and constraints of the problem.The problem inequality constraints (constraints 3-21) are related to product specifications and safety or performance limits. The equality constraints 1 and 2 were included to model the heat integration between the atmospheric column and the feed pre-heating train. Another 18 process variables take part of the objective function, as... 

Integral constraints could be equality or inequality constraints. We first consider integral equality constraints in an optimal control problem with free state and free final time. 

This problem is similar to that in Section 7.2.4 (p. 214) except that the integral equality constraints are replaced with the inequalities... 

Path constraints g represent conditions that must be fulfilled throughout the entire integration horizon. These inequality constraints augment the algebraic equations ft. 

Indirect or variational approaches are based on Pontryagin s maximum principle [8], in which the first-order optimality conditions are derived by applying calculus of variations. For problems without inequality constraints, the optimality conditions can be written as a set of DAEs and solved as a two-point boundary value problem. If there are inequality path constraints, additional optimality conditions are required, and the determination of entry and exit points for active constraints along the integration horizon renders a combinatorial problem, which is generally hard to solve. There are several developments and implementations of indirect methods, including [9] and [10]. 

The diffusion matrix can be estimated from a single experiment with good precision using the incremental approach. It should be noted that the four diffusion coefficients are not identifiable from Eq. (5). But the insertion of the constitutive law for the diffusion coefficient (Eq. (3)) into the flux expression allows to overcome this situation. Purthermore, it should be stressed that this estimation problem is very difficult to solve by the simultaneous approach. The Pick matrix is positive definite which is enforced by three inequality constraints (Taylor and Krishna, 1993). In parameter estimation, a sequential approach with an infeasible path optimization routine is often used. This may not be possible since the model cannot be integrated if the matrix is not positive definite. This limitation does not apply to the incremental approach since no solution of the direct problem is required. It could therefore also be used to initialize the simultaneous procedure. 

The convexity of the exponential functions then implies the inequality (Q) > 0, which resembles the second law. The integral fluctuation theorem implies that there are trajectories for which Q is negative with the exception of the degenerate case, p(Q) = d(Q) leading to violation of the second law. One constraint on the probability distribution p(Q). If p(Q) is a Gaussian, the integral flucmation theorem implies the relation ((Q — (O)) ) = 2(Q) between the variance and the mean of Q (Seifert, 2012). 

Point constraints correspond to conditions that must be fulfilled at a certain point in the integration horizon they can be either equalities (ftp) or inequalities (gp). They are referred to as end point constraints if they are imposed at the end of the horizon tf (a typical example in chemical engineering is a desired product concentration at the end of the operation in a batch process). If these additional conditions are imposed at any other point in the horizon, they are called interior-point constraints. 

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